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Temporal convergence of a locally implicit discontinuous Galerkin method for Maxwell’s equations

Ludovic Moya (2012)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

In this paper we study the temporal convergence of a locally implicit discontinuous Galerkin method for the time-domain Maxwell’s equations modeling electromagnetic waves propagation. Particularly, we wonder whether the method retains its second-order ordinary differential equation (ODE) convergence under stable simultaneous space-time grid refinement towards the true partial differential equation (PDE) solution. This is not a priori clear due to the component splitting which can introduce order...

Temporal convergence of a locally implicit discontinuous Galerkin method for Maxwell’s equations∗

Ludovic Moya (2012)

ESAIM: Mathematical Modelling and Numerical Analysis

In this paper we study the temporal convergence of a locally implicit discontinuous Galerkin method for the time-domain Maxwell’s equations modeling electromagnetic waves propagation. Particularly, we wonder whether the method retains its second-order ordinary differential equation (ODE) convergence under stable simultaneous space-time grid refinement towards the true partial differential equation (PDE) solution. This is not a priori clear due to the component splitting which can introduce order...

The Back and Forth Nudging algorithm for data assimilation problems : theoretical results on transport equations

Didier Auroux, Maëlle Nodet (2012)

ESAIM: Control, Optimisation and Calculus of Variations

In this paper, we consider the back and forth nudging algorithm that has been introduced for data assimilation purposes. It consists of iteratively and alternately solving forward and backward in time the model equation, with a feedback term to the observations. We consider the case of 1-dimensional transport equations, either viscous or inviscid, linear or not (Burgers’ equation). Our aim is to prove some theoretical results on the convergence,...

The Back and Forth Nudging algorithm for data assimilation problems : theoretical results on transport equations

Didier Auroux, Maëlle Nodet (2012)

ESAIM: Control, Optimisation and Calculus of Variations

In this paper, we consider the back and forth nudging algorithm that has been introduced for data assimilation purposes. It consists of iteratively and alternately solving forward and backward in time the model equation, with a feedback term to the observations. We consider the case of 1-dimensional transport equations, either viscous or inviscid, linear or not (Burgers’ equation). Our aim is to prove some theoretical results on the convergence, and convergence properties, of this algorithm. We...

The Back and Forth Nudging algorithm for data assimilation problems : theoretical results on transport equations

Didier Auroux, Maëlle Nodet (2012)

ESAIM: Control, Optimisation and Calculus of Variations

In this paper, we consider the back and forth nudging algorithm that has been introduced for data assimilation purposes. It consists of iteratively and alternately solving forward and backward in time the model equation, with a feedback term to the observations. We consider the case of 1-dimensional transport equations, either viscous or inviscid, linear or not (Burgers’ equation). Our aim is to prove some theoretical results on the convergence,...

The correct use of the Lax–Friedrichs method

Michael Breuß (2004)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We are concerned with the structure of the operator corresponding to the Lax–Friedrichs method. At first, the phenomenae which may arise by the naive use of the Lax–Friedrichs scheme are analyzed. In particular, it turns out that the correct definition of the method has to include the details of the discretization of the initial condition and the computational domain. Based on the results of the discussion, we give a recipe that ensures that the number of extrema within the discretized version of...

The correct use of the Lax–Friedrichs method

Michael Breuß (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

We are concerned with the structure of the operator corresponding to the Lax–Friedrichs method. At first, the phenomenae which may arise by the naive use of the Lax–Friedrichs scheme are analyzed. In particular, it turns out that the correct definition of the method has to include the details of the discretization of the initial condition and the computational domain. Based on the results of the discussion, we give a recipe that ensures that the number of extrema within the discretized version...

The fourth order accuracy decomposition scheme for an evolution problem

Zurab Gegechkori, Jemal Rogava, Mikheil Tsiklauri (2004)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

In the present work, the symmetrized sequential-parallel decomposition method with the fourth order accuracy for the solution of Cauchy abstract problem with an operator under a split form is presented. The fourth order accuracy is reached by introducing a complex coefficient with the positive real part. For the considered scheme, the explicit a priori estimate is obtained.

The fourth order accuracy decomposition scheme for an evolution problem

Zurab Gegechkori, Jemal Rogava, Mikheil Tsiklauri (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

In the present work, the symmetrized sequential-parallel decomposition method with the fourth order accuracy for the solution of Cauchy abstract problem with an operator under a split form is presented. The fourth order accuracy is reached by introducing a complex coefficient with the positive real part. For the considered scheme, the explicit a priori estimate is obtained.

The numerical interface coupling of nonlinear hyperbolic systems of conservation laws : II. The case of systems

Edwige Godlewski, Kim-Claire Le Thanh, Pierre-Arnaud Raviart (2005)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We study the theoretical and numerical coupling of two hyperbolic systems of conservation laws at a fixed interface. As already proven in the scalar case, the coupling preserves in a weak sense the continuity of the solution at the interface without imposing the overall conservativity of the coupled model. We develop a detailed analysis of the coupling in the linear case. In the nonlinear case, we either use a linearized approach or a coupling method based on the solution of a Riemann problem. We...

The numerical interface coupling of nonlinear hyperbolic systems of conservation laws: II. The case of systems

Edwige Godlewski, Kim-Claire Le Thanh, Pierre-Arnaud Raviart (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

We study the theoretical and numerical coupling of two hyperbolic systems of conservation laws at a fixed interface. As already proven in the scalar case, the coupling preserves in a weak sense the continuity of the solution at the interface without imposing the overall conservativity of the coupled model. We develop a detailed analysis of the coupling in the linear case. In the nonlinear case, we either use a linearized approach or a coupling method based on the solution of a Riemann problem....

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